Arc length
Problem 5.199 · medium
Find the length of the curve \( \displaystyle y = \frac{2 x^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 1 \) to \( \displaystyle x = 8 \).
- Arc length is ∫ √(1 + (dy/dx)²) dx.Reviewed
- \[ \frac{d}{d x} \frac{2 x^{\frac{3}{2}}}{3} = \sqrt{x} \]dy/dx.✓ Proved
- \[ x + 1 \]1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
- \[ \int\limits_{1}^{8} \sqrt{x + 1}\, dx = 18 - \frac{4 \sqrt{2}}{3} \]Integrate.✓ Proved
Answer \( 18 - \frac{4 \sqrt{2}}{3} \)
Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | a 200,000-segment polygon along the curve has the same length |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution incorrectly simplifies 1 + (dy/dx)^2 to x + 1, whereas it should be 1 + x. This leads to an incorrect integral and final answer.gpt-oss:20b: pass 2026-10-03
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/arc_length, checked 2026-10-03 with SymPy 1.14.0.